On the Liouville theorem for harmonic maps
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- by Hyeong In Choi PDF
- Proc. Amer. Math. Soc. 85 (1982), 91-94 Request permission
Abstract:
Suppose $M$ and $N$ are complete Riemannian manifolds; $M$ with Ricci curvature bounded below by $- A$, $A \geqslant 0$, $N$ with sectional curvature bounded above by a positive constant $K$. Let $u:M \to N$ be a harmonic map such that $u(M) \subset {B_R}({y_0})$. If ${B_R}({y_0})$ lies inside the cut locus of ${y_0}$ and $R < \pi /2\sqrt K$, then the energy density $e(u)$ of $u$ is bounded by a constant depending only on $A$, $K$ and $R$. If $A = 0$, then $u$ is a constant map.References
- Shiu Yuen Cheng, Liouville theorem for harmonic maps, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 147–151. MR 573431
- R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983
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Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 85 (1982), 91-94
- MSC: Primary 53C99; Secondary 58E20
- DOI: https://doi.org/10.1090/S0002-9939-1982-0647905-3
- MathSciNet review: 647905